ANalysis Of VAriance

Data Visualization Principles

Minimalism

One of Tufte’s ideas is data-to-ink ratio, generally stating that when creating a visualization, you should try to maximize the amount of “ink” (physical or digital) used to represent the data, and minimize the amount used to represent everything else not important to understanding.

theme_classic() in ggplot()

Gestalt Principles

Gestalt psychology is a theory of perception that believes humans are inclined to understand objects as an entire structure rather than the sum of its parts.

Gestalt Principles for Data Visualization

The Figure and Ground Principle describes the capacity to perceive the relationship between form and surrounding space to create meaning. A sense of wholeness or unity depends on how you perceive the relationship between an object and the area in which it is contained. The ‘figure’ is the focus element, while the ‘ground’ is the figure’s background.

theme_bw() in ggplot()

List of all ggplot() themes

Moving into ANOVA

Before…

Now…

Goal of an ANOVA

Analysis of variance (ANOVA) compares the means of three of more groups to detect if the means of the groups are different.

How???

Compare how different a group of means are
Scale the differences relative to the variability of the groups
Summarize the differences with one number

Visualizing Group Differences

We want visualizations that allow for us to easily compare:

the center (mean) of the groups
the spread (variability) of the groups

What can you say about the differences between the groups?

What can you say about the variability within the groups?

Carrying out an ANOVA

Step 1: Compare your groups

Step 2: Find the overall mean

This ignores the groups and finds one mean for every observation!

Step 3: Find the group means

Step 4: Calculate the sum of squares

Step 5: Calculate the F-statistic

Can an F-statistic be negative?

Step 6: Find the p-value

F-distribution

An \(F\)-distribution is a variant of the \(t\)-distribution, and is also defined by degrees of freedom.

This distribution is defined by two different degrees of freedom:

from the numerator (MSG) : \(k - 1\)
from the denominator (MSE) : \(n - k\)

Two degrees of freedom!

Changing the numerator degrees of freedom

Changing the denominator degrees of freedom

Do you always use an F-distribution to get the p-value?

NO!

Conditions of an ANOVA

Independence
- Within groups
- Between groups
Normality of the responses
- The distribution of each group is approximately normal
Equal variability of the groups
- The spread of the distributions are similar across groups

What do you think?

If the normality condition is violated what type of method should we use?

Simulation-based Methods

Step 1: Calculating the Observed F-statistic

obs_F <- evals_small %>% 
  specify(response = min_eval, 
          explanatory = age_cat) %>% 
  calculate(stat = "F")

Response: min_eval (numeric)
Explanatory: age_cat (factor)
# A tibble: 1 × 1
   stat
  <dbl>
1  1.41

Step 2: Simulating what could have happened under \(H_0\)

How could we use cards to simulate what minimum evaluation score a professor would have gotten, if their score was independent from their age?

Another Permutation Distribution

null_dist <- evals_small %>% 
  specify(response = min_eval, 
          explanatory = age_cat) %>% 
  hypothesize(null = "independence") %>% 
  generate(reps = 2000, type = "permute") %>% 
  calculate(stat = "F")

Another Permutation Distribution

Why doesn’t the distribution have negative numbers?

Visualizing the p-value

visualise(null_dist) + 
  shade_p_value(obs_stat = obs_F, direction = "greater")

What would you conclude regarding the mean minimum evaluation score and different age groups of professors?

Theory-based Methods

Using aov()

aov(min_eval ~ age_cat, 
    data = evals_small)  %>% 
  tidy()

# A tibble: 2 × 6
  term         df sumsq meansq statistic p.value
  <chr>     <dbl> <dbl>  <dbl>     <dbl>   <dbl>
1 age_cat       3  1.24  0.414      1.41   0.244
2 Residuals    90 26.4   0.293     NA     NA

How was the statistic calculated?

What distribution was used to calculate the p.value?

What would you conclude regarding the mean minimum evaluation score and different age groups of professors?

Did the two methods yield different results?

Next…